Currently there is no reliable method for testing if a browser is in private browsing mode. See this stack overflow answer for more details. The whole point of private mode is that once the tab or session is closed, all data is erased, trying to circumvent this will only give your problems.

For my own experience I think that any person that uses private browsing mode on a regular basis is aware that nothing is saved and so would not expect sessions to persist between uses without logging in.